Constrained Online Convex Optimization without Slater's Condition

2026-06-30

AI Summary

This work addresses the challenge of simultaneously handling adversarial losses and stochastic or adversarial constraints in online convex optimization without requiring the Slater condition. The authors propose an anytime primal-dual framework that stabilizes the dual dynamics by incorporating an adaptive regularization term in the dual updates. This approach is the first to achieve near-optimal regret and constraint violation bounds without assuming the Slater condition, while also providing high-probability guarantees. Specifically, for stochastic constraints and convex losses, the method attains an expected regret of $O(\sqrt{T})$ and expected cumulative constraint violation of $O(\sqrt{T}\log T)$. In the strongly convex setting, the regret improves to $O(\log T)$, and the framework naturally extends to scenarios with adversarial constraints.

Abstract

We study constrained online convex optimization with adversarial losses and stochastic or adversarial constraints. For stochastic constraints, existing algorithms that achieve nearly optimal regret and constraint violation bounds typically rely on regularity assumptions such as Slater's condition, while adversarial-constraint algorithms avoid these assumptions by using a rather restrictive round-wise feasible comparator. We bridge this gap with an anytime primal-dual framework that incorporates an adaptive regularizer into the dual update. The regularizer stabilizes the dual process without relying on the negative drift induced by Slater's condition. For stochastic constraints and convex losses, our algorithm achieves $O(\sqrt{T})$ expected regret and $O(\sqrt{T}\log T)$ expected cumulative constraint violation. Furthermore, we show that our algorithm also admits high-probability bounds of the same order on regret and constraint violation. For strongly convex losses, the regret bound improves to $O(\log T)$ with a violation bound of the same order. With a minor modification, the framework also applies to adversarial constraints and provides guarantees for hard constraint violation.

Problem

• Constrained Online Convex Optimization
• Slater's Condition
• Adversarial Constraints
• Stochastic Constraints
• Regret and Constraint Violation

Innovation

• Online Convex Optimization
• Primal-Dual Framework
• Adaptive Regularization
• Constraint Violation
• Slater's Condition